Low-Power Magnetoplasmadynamic Thruster Numerical Performance Model

Abstract

Magnetoplasmadynamic thrusters represent a promising Electric Propulsion technology for future space missions; however, their optimization is hampered by the lack of accurate performance models in the emerging regime of low power (<12 kW) and high magnetic fields (>0.1 T), where traditional formulations prove inadequate. In this work, a new semi-empirical model for predicting the thrust and discharge voltage of argon-fed MPD thrusters was developed and validated. Starting from state-of-the-art physical models, multi-factorial correction factors were introduced to account for the coupled effects of discharge current (8–180 A), mass flow rate (3–21 mg/s), and applied magnetic field (up to 0.6 T). The model was calibrated and validated using a comprehensive and homogeneous collection of experimental data from the literature. A comparative analysis demonstrates that the corrected model significantly reduces prediction errors (0–9%) compared to reference models available in the literature (8–50%). In particular, the model exhibits remarkably superior accuracy in both the Self-Field and Applied-Field regimes, overcoming the main limitations of previous formulations and providing more robust estimates across a wide operational envelope. The developed model constitutes a reliable and physically consistent tool for the analysis and preliminary design of low-power, argon-fed magnetoplasmadynamic thrusters, enabling more effective optimization for this class of propulsion systems.

1. Introduction

The growing demand for more efficient and long-duration space missions has spurred a profound interest in Electric Propulsion (EP) technologies. Unlike chemical propulsion, EP systems use energy from an external source to accelerate a propellant to extremely high exhaust velocities. This results in a significantly higher specific impulse (Isp), which drastically reduces the required propellant mass and is crucial for missions such as orbital transfers and interplanetary exploration. Within the landscape of electric propulsion, the category of electromagnetic systems is particularly promising for applications requiring high thrust density [1].

The operation of Magnetoplasmadynamic Thrusters (MPDTs) is based on the generation and acceleration of plasma within a coaxial chamber composed of an annular anode and a central cylindrical cathode. A high potential difference applied between the electrodes initiates a discharge that ionizes the propellant [2].

The plasma thus created is accelerated through the interaction between the electric current and the magnetic field, in accordance with the Lorentz law [3].

Two main MPDT configurations can be identified:

• Self-Field MPDTs (SF-MPDTs). In this configuration, thrust is generated by the interaction between the discharge current and the self-induced magnetic field. Specifically, the current induces an azimuthal magnetic field ${\overrightarrow{B}}_{\theta }$, and its interaction with the radial current density component (${\overrightarrow{j}}_r$) produces a Lorentz force. The axial component of this interaction (${\overrightarrow{j}}_r\times {\overrightarrow{B}}_{\theta }$), called the “blowing force“, is responsible for the acceleration. Although Figure 1 depicts an applied-field thruster, this self-field component is present and indicated by the vector ${\overrightarrow{j}}_r\times {\overrightarrow{B}}_{\theta }$.
• Applied-Field MPDTs (AF-MPDTs). In this configuration, shown in Figure 1, the addition of an external solenoid generates a strong applied magnetic field ($B_a$) directed along the thruster axis. The radial component of the discharge current density (${\overrightarrow{j}}_r$) interacts with the axial magnetic field ($B_z\approx B_a$), creating an azimuthal Lorentz force (${\overrightarrow{j}}_r\times {\overrightarrow{B}}_z$) that induces a swirling motion of the plasma. This mechanism is known as “swirl acceleration“, where the swirling plasma expands within the “magnetic nozzle” formed by the applied field to generate axial thrust. Furthermore, the swirling motion induces an azimuthal Hall current, whose interaction with the applied field generates another significant thrust component (${\overrightarrow{j}}_{\theta }\times {\overrightarrow{B}}_z$), called “Hall acceleration”.

In addition to the electromagnetic contribution, there is also a gasdynamic component linked to the expansion of the plasma heated by the discharge. This contribution, however, becomes negligible at high power levels [4], where acceleration is dominated by the described electromagnetic phenomena.

Within the electromagnetic category, magnetoplasmadynamic thrusters, and particularly their Applied-Field configuration (AF-MPDT), stand out for their ability to operate at significantly higher power levels and thrust densities compared to other EP systems [2]. Furthermore, the external magnetic field enhances both thrust and efficiency [5], while also ensuring greater operational stability, especially at lower current levels where the self-field component alone would be insufficient.

Recent technological advancements, particularly in the development of High-Temperature Superconductors (HTS), have renewed interest in a previously underexplored operational regime for AF-MPDTs: the low-power (typically < 50 kW) and high-magnetic-field regime [6,7]. This new operational scenario is particularly promising for satellites with limited power availability but poses a significant modeling challenge. Previous performance models, developed and calibrated for high-power regimes (>100 kW), have shown significant predictive limitations when applied to these new operating conditions. As highlighted in recent reviews [6], these discrepancies underscore the critical need to develop more accurate predictive tools.

The scientific literature offers several semi-empirical models for estimating thrust and discharge voltage [4,8,9,10,11,12]. For thrust, formulations range from the classic ones, such as Maecker’s model [13] for the self-field component, to more recent and complex correlations like that of Coogan et al. [4] for the applied-field component. For voltage, models are often based on the decomposition into fundamental energy contributions, as proposed by Lev [12]. However, even the most recent models, such as that by Balkenhohl et al. [6], which represents the state-of-the-art for the low-power regime, rely on purely statistical corrections that, despite improving accuracy, do not fully capture the complex interdependence among operational parameters. Furthermore, many of these models attempt to be universal by including data from different propellants (e.g., argon, lithium, hydrogen), which inevitably increases predictive uncertainty due to the different physicochemical properties of the generated plasma.

To overcome these limitations, the development and validation methodology of an improved performance model, specifically designed for MPD thrusters operating with argon propellant in the low-power regime, is presented in Section 2. The key innovation of this work lies in the introduction of multi-factorial correction factors, which are selectively applied to the most critical components of the baseline physical models to account for the coupled effects of discharge current, mass flow rate, and magnetic field, as detailed in Section 4 and Section 5. The experimental database used for parameter regression and model validation is described in Section 3. Finally, Section 6 presents the results of the validation, demonstrating the model’s capability to provide an accurate and robust design tool, followed by a discussion and concluding remarks in Section 7 and Section 8.

2. Performance Model Development Methodology

Here, the numerical model for thrust and discharge voltage building on recent developments in the literature is presented. The goal is to quantify how the key operating parameters, discharge current $I_d$, mass flow rate $\dot{m}$, and applied magnetic field $B_a$, affect the performance of a MPDT operating in a low-power (<12 kW) and high-mass-flow (3–21 mg/s) regime, in terms of thrust T, specific impulse $I_{sp}$, and efficiency $\eta$.

Flowchart of the Model

The methodological foundation for the performance model developed in this work is the semi-empirical Low-Power (LP) model proposed by Balkenhohl et al. [6]. This approach represents a significant step forward, with respect to the previous model available in literature [4,8,9,10], as it combines a solid physical basis, derived from fundamental models such as Coogan et al. (2017) [4], with an extensive statistical analysis of experimental data. Balkenhohl’s model [6] was specifically conceived to overcome the limitations of classical formulations, which are typically calibrated for high-power regimes (>100 $\mathrm{kW}$) [6], and to provide more reliable predictions for the new generation of thrusters operating in the low-kilowatt range.

However, preliminary analysis highlighted that, although Balkenhohl’s model [6] provides an excellent baseline, there remain opportunities for great improvement, particularly in predicting performance trends under the simultaneous variation of discharge current, applied magnetic field, and propellant mass flow rate.

In this work, the model has therefore been further developed through the introduction of a multifactorial correction term applied selectively to the most critical components of the physical description and precisely:

• For the thrust model, the correction was applied to the applied-field electromagnetic thrust component ($T_{af}$) yielding to the corrected variable ($T_{corr}$). This component was selected because it proved to be the most sensitive to changes in the operating parameters.
• For the voltage model, an analogous approach was taken. Starting from the corrected thrust $T_{corr}$, the estimation of the back-electromotive force was further refined to obtain ($V_{corr}$) by introducing a non-linear correction factor that accounts for the interactions between current $I_d$, mass flow rate $\dot{m}$, and magnetic field $B_a$.

The model developed here therefore consists of a new and more reliable way to estimate the two fundamental physical quantities of the thruster, namely the total thrust generated, $T_{corr}$, and the discharge voltage required, $V_{corr}$.

From these variables, all the other performance metrics of interest are derived, including the specific impulse $I_{sp}$ and the overall efficiency $\eta$. The developed model has been implemented following a modular structure, in order to make clear the logical flow that connects the input parameters to the performance outputs of interest. Figure 2 shows in a detailed flowchart the internal structure of the model. The thrust calculation begins with the determination of three contributions: the gasdynamic component $T_{gd}$ (Equation (2)), the self-field component $T_{sf}$ (Equation (3)), and the applied-field component $T_{af}$ (Equation (4)). The latter is corrected with the new multi-variable factor $F_T$, developed in this work, which accounts for the dependence of thrust on discharge current, mass flow rate, and applied magnetic field. Once the applied-field component is corrected, the three contributions are summed to obtain a preliminary total thrust, which is then further modified through the Balkenhohl correction (Equation (5)), which depends on the magnetic field. The result of this sequence is the corrected thrust, indicated as $T_{corr}$.

The voltage model then uses the corrected thrust ($T_{corr}$) as its starting point. From this, the electromotive voltage $V_e$, is derived and combined with the other physical contributions, namely electrode voltage drops and the energetic costs of ionization and plasma heating. The total voltage thus obtained is further corrected by a second multivariable factor $F_V$, which accounts for the combined dependence of voltage on discharge current, mass flow rate, applied magnetic field, and anode radius. The outcome is the corrected discharge voltage, indicated as $V_{corr}$.

From these two fundamental physical quantities, namely the total thrust generated and the discharge voltage required, all other key quantities are derived. These include the primary performance indicators, such as specific impulse $I_{sp}$ and overall efficiency ${\eta }_T$. Additionally, the model calculates thermal losses $Q_{th}$ and the resulting anode and cathode temperatures $T_{anode}$ and $T_{cathode}$, as well as the onset current $J_{onset}$ for stability assessment, which are crucial for engineering design.

In the following Section 4 and Section 5, the procedure used to derive the corrective factors $F_T$ and $F_V$, for thrust and voltage respectively, will be described in detail, and it will be shown how their introduction significantly improves the predictive capability of the model compared to available experimental data that are presented next in Section 3.

3. Data Collection

To estimate the values and validate the model of thrust and voltage, an argon-consistent experimental dataset for low–power MPDTs has been assembled from published literature. Where the original sources also report tests with other gases, these are mentioned only as context; the validation relies exclusively on measurements with argon. Data selection is limited to thrusters with power $P<12$ kW and relatively high applied magnetic fields up to $B_a\simeq 0.6$ T.

Table 1. Low–power AF–MPDTs used for validation (argon data).

Thruster	Power [kW]	${\mathit{I}}_{\mathit{d}}$ [A]	${\mathit{B}}_{\mathit{a}}$ [mT]	Cathode Type	Propellant	Ref.
DFVLR (1975)	7.0–11.6	80	∼600	rod	Ar, Kr, Xe	[14]
Waseda Univ. (2003)	∼0.5–1.1	8–36	100–200	rod	Ar, Ne, He	[15]
Beihang Univ. (2018)	∼10	90–180	50–170	n.a.	Ar	[16]
Beihang Univ. (2019)	5–10	100–180	0–133	hollow	Ar	[17]

Notes. For model calibration/validation only argon data have been used; where sources report other gases (Ne, He, Kr, Xe), those results are cited for context but not used in the fit. n.a. = not available.

lists, in chronological order, the main thrusters that fall within this range of interest.

The DFVLR research group in Stuttgart developed the X16 thruster in the 1970s, as the culmination of a long AF–MPDT program. The device operated between 7 and 12 kW and was tested with several propellants (argon, krypton, and xenon), delivering thrust on the order of 200–250 mN. Tests were performed at 7–9 mg/s mass flow and an applied magnetic field up to 0.6 T, demonstrating continuous, stable operation for over 100 h. The anode temperature is also reported (≈1850–2200 °C). These data provide a valuable reference on how high $B_a$ values and a moderate mass flow influence performance and electrode loads [14].

At Waseda University (Japan), an ∼1 kW AF–MPD arc-jet was developed and tested mainly with argon, but also with neon and helium. The device was operated at a very low mass flow (0.2–5 mg/s) and applied fields of up to 0.2 T. Despite stability challenges in this regime, a specific impulse up to 2200 s and a maximum efficiency of 8.5% were achieved [15].

The Beihang University campaign investigated the high-flow/low-current regime on two configurations (cylindrical and convergent–divergent anodes) by fixing $\dot{m}=21$ mg/s of argon and conducting tests both without $B_a$ and with $B_a$ up to ≃0.133 T (measured at the cathode tip). The results show that increasing $B_a$ and increasing the cathode injection fraction improve performance, with a more pronounced benefit for the cylindrical anode; in this regime, discharge currents are on the order of 100–180 A [17].

Finally, a subsequent Beihang study introduced the concept of the effective voltage and demonstrated its experimental impact on a ∼10 kW AF–MPDT, exploring $I_d=90–180$ A and $B_a=0.05–0.17$ T. The analysis indicates that increasing the fraction of voltage effectively converted into acceleration (also via the management of $\dot{m}$ and the cathode fraction) is a direct route to improve the efficiency [16].

This dataset, representative of the regimes of interest, enables a comparable quantification of thrust and voltage (and the derived parameters specific impulse and efficiency) versus $I_d$, $\dot{m}$, and $B_a$, providing the necessary reference for the development and validation of the model.

4. Development of the Improved Thrust Model

An in-depth analysis of the Balkenhohl model [6], which will be discussed in detail in the upcoming Section 6, showed that its predictions are not always in good agreement with experimental data, revealing significant discrepancies in some cases. The primary source of this error was identified in the modeling of the applied-field thrust component, $T_{af}$, which was found to be particularly sensitive to the operating conditions.

To overcome this limitation, the general structure of the Balkenhohl model [6] was retained, while introducing a more targeted correction approach. Specifically, a multiplicative factor, $F_T$, was defined and applied exclusively to the $T_{af}$ component, with the goal of reducing experimental discrepancies without altering the other contributions. Under these conditions, the total thrust of the corrected model, $T_{corr}$, is expressed as:

$$T_{corr}={\displaystyle \frac{T_{gd}+T_{sf}+F_T·T_{af}}{1+\frac{\gamma }{100}}}$$

(1)

According to Tikhonov [18], the gasdynamic thrust term can be calculated as:

$$T_{gd}=k_{GD}\dot{m}{a}_0$$

(2)

where $a_0$ is the ion sound velocity, $\dot{m}$ the mass flow rate, and $k_{GD}$ a dimensionless coefficient depending on the propellant type and thruster geometry that can be approximated to unity [6,9].

The second term, namely the self-field component $T_{sf}$, given by Maecker’s [13] formulation (Equation (3)), results to be:

$$T_{sf}={\displaystyle \frac{{\mu }_0}{4\pi }}\left[ln\left({\displaystyle \frac{r_a}{r_c}}\right)+{\displaystyle \frac{3}{4}}\right]I_d^2$$

(3)

The latter term, the applied-field component $T_{af}$, calculated through Coogan’s [4] model (Equation (4)), is corrected with the new multi-variable factor $F_T$ which accounts for the dependence of thrust on discharge current $I_d$, mass flow rate $\dot{m}$, and applied magnetic field $B_a$ as in Equation (7). Here, $r_a$ and $r_c$ are the anode and cathode radius, respectively.

$$T_{af}=1.14I_d{B}_a{r}_a{\varphi }^{-0.13}{\left({\displaystyle \frac{r_a}{r_c}}\right)}^{-0.3}{\left(10+{\displaystyle \frac{l_c}{l_a}}\right)}^{-0.67}$$

(4)

where $\varphi$ is a factor that describes the alignment between the anode and the magnetic field lines, defined in Coogan’s model [4], and $l_a$ and $l_c$ are the anode and cathode length, respectively. Once the applied-field component is corrected, the three contributions are summed to obtain a preliminary total thrust, which is then further modified through the Balkenhohl correction [6] $(1+\gamma /100)$:

$$\gamma \left(B_a\right)=102.4B_a-41.07$$

(5)

which depends on the magnetic field $B_a$.

At the end of the whole sequence, the corrected thrust, indicated as $T_{corr}$, is obtained.

The derivation of an analytical formula for $F_T$ followed a semi-empirical path based on experimental data, which involved several phases.

The first step consisted of mathematically isolating the factor. By enforcing the condition that the thrust model is equal to the experimentally measured thrust ($T_{exp}$), Equation (1) was inverted to express $F_T$ as a function of known quantities. This yields:

$$F_{T,exp}={\displaystyle \frac{T_{exp}\left(1+\frac{\gamma }{100}\right)-(T_{gd}+T_{sf})}{T_{af}}}$$

(6)

This relationship allowed for the calculation of a reference value, $F_{T,exp}$, for each experimental point, thus creating a dataset that represents the true behavior of the thruster.

Subsequently, a power law functional relationship was assumed between $F_T$ and the primary operating variables: discharge current ($I_d$), mass flow rate ($\dot{m}$), and applied magnetic field ($B_a$). This choice is motivated by the fact that such a form is often suitable for describing complex physical phenomena:

$$F_T=C_T·I_d^{{\alpha }_T}·{\dot{m}}^{{\beta }_T}·B_a^{{\delta }_T}$$

(7)

To estimate the coefficients of this model, the power law was transformed into a linear form, which is simpler to analyze. By applying the logarithm, the equation becomes:

$$\log \left(F_T\right)=\log \left(C_T\right)+{\alpha }_T\log \left(I_d\right)+{\beta }_T\log \left(\dot{m}\right)+{\delta }_T\log \left(B_a\right)$$

(8)

In this form, the problem is reduced to a multivariable linear regression, where $\log \left(F_T\right)$ is the dependent variable and the logarithms of the operating quantities are the independent variables. The analysis was conducted in the MATLAB^® (R2025b) environment [19] using the least-squares method, which allowed for the determination of the constants $C_T$, ${\alpha }_T$, ${\beta }_T$, and ${\delta }_T$.

Finally, to ensure a compact form and to render the coefficients dimensionless, a normalization procedure was applied. Each variable was divided by a reference value chosen in accordance with the operating domain:

$$I_{d,ref}=100\mathrm{A},{\dot{m}}_{ref}=10\mathrm{mg}/\mathrm{s},B_{a,ref}=0.1\mathrm{T}$$

The final form of the model for the correction factor is therefore:

$$F_T=C_T·{\left({\displaystyle \frac{I_d}{I_{d,ref}}}\right)}^{{\alpha }_T}·{\left({\displaystyle \frac{\dot{m}}{{\dot{m}}_{ref}}}\right)}^{{\beta }_T}·{\left({\displaystyle \frac{B_a}{B_{a,ref}}}\right)}^{{\delta }_T}$$

(9)

where the coefficients estimated from the regression are reported in the first row of

Table 2. Thrust model coefficients and validation.

	${\mathit{C}}_{\mathit{T}}$	${\mathit{\alpha }}_{\mathit{T}}$	${\mathit{\beta }}_{\mathit{T}}$	${\mathit{\delta }}_{\mathit{T}}$
Coefficient	0.51	0.77	1.00	1.10
Valid. Coeff.	0.51	0.77	1.00	1.10

as follows. In order to establish the robustness of the method and the sensitivity of the coefficients, other two thruster configurations have been added to the dataset, generating a new set of coefficients that is completely unaltered, demonstrating its reliability, as can be seen in the second row of Table 2.

This allows for the reduction of discrepancies between the theoretical model and experimental data, improving the predictive capability compared to the correction originally proposed by Balkenhohl [6], which was based solely on the applied magnetic field.

5. Development of the Improved Voltage Model

Unlike the thrust model, for which an existing architecture was improved, the development of the voltage model followed an alternative approach, based on a different physical foundation than that proposed by Balkenhohl et al. [6].

The starting point is Lev’s theoretical model [12], a fundamental reference that decomposes the total discharge voltage into the sum of its physical contributions as in Equation (10).

$$V_{\mathrm{Lev}}=V_{\mathrm{e}}+V_{\mathrm{heat}}+V_i+{\mathsf{\Phi }}_a+{\mathsf{\Phi }}_c+V_a$$

(10)

Here, all the terms have a specific physical meaning:

• $V_{\mathrm{e}}$ (Back-Electromotive Voltage) is the “useful” component of voltage, directly associated with the kinetic power imparted to the plasma jet and hence with thrust production. Often referred to as the back electromotive voltage, it is expressed as:

  $$V_{\mathrm{e}}={\displaystyle \frac{T^2}{2\dot{m}{I}_d}}$$

  (11)
• $V_i$ (Ionization Voltage) represents the potential drop corresponding to the energetic cost of ionizing the neutral propellant. It depends on the first ionization energy of the gas (${\epsilon }_i$) and the mass flow rate:

  $$V_i={\displaystyle \frac{\dot{m}}{m_i{I}_d}}{\epsilon }_i$$

  (12)
• $V_{\mathrm{heat}}$ (Plasma Heating Voltage) quantifies the power dissipated in raising the temperature of the ions ($T_i$) and electrons ($T_e$), which does not directly contribute to thrust. It is calculated as:

  $$V_{\mathrm{heat}}={\displaystyle \frac{\dot{m}}{m_i{I}_d}}{k}_B(T_e+T_i)$$

  (13)

  where $k_B$ is the Boltzmann constant.
• $V_a$ (Anode Sheath Voltage) models the potential drop across the anode sheath. In this work, the sheath voltage drop is specified solely for the anode and involves a semi-empirical formulation based on the balance between random electron flux and thermionic emission from the anode surface:

  $$V_a={\displaystyle \frac{k_B{T}_e}{e}}ln\left({\displaystyle \frac{\frac{I_d}{A_a}+A_R{T}_a^{2}exp\left(-\frac{e{\mathsf{\Phi }}_a}{k_B{T}_a}\right)}{\frac{1}{4}e{n}_e\sqrt{\frac{8k_B{T}_e}{\pi m_e}}}}\right)$$

  (14)

  where e is the elementary charge, $A_a$ the anode surface area, $A_R$ the Richardson constant, $n_e$ the electron density, $m_e$ the electron mass, and ${\mathsf{\Phi }}_a$ the anode work function. The definitions and auxiliary relations used to evaluate these variables are adopted from the work of Coogan et al. [4].
• ${\mathsf{\Phi }}_a$ and ${\mathsf{\Phi }}_c$ (Work Functions) represent the minimum energy required to extract electrons from the anode and cathode surfaces, respectively.

The main innovation in this work is to integrate this model with the results obtained for the thrust: the back-electromotive voltage term ($V_e$), the dominant component of the voltage, was calculated using the corrected thrust $T_{corr}$ as:

$$V_{\mathrm{e},corr}={\displaystyle \frac{T_{corr}^2}{2\dot{m}{I}_d}}$$

(15)

This crucial step creates a direct and physically coherent link between the thrust and voltage models, leading to the definition of a model’s base voltage, $V_{Lev}$ (Equation (10)), which represents the best physical estimate of the discharge voltage before any further correction:

$$V_d=V_{e,corr}+V_i+V_{heat}+V_a+{\mathsf{\Phi }}_a+{\mathsf{\Phi }}_c$$

(16)

Therefore, it was decided not to use Balkenhohl’s $\beta \left(r_a\right)$ factor and to develop a new multi-variable correction factor, $F_V$.

The final model for the corrected voltage, $V_{corr}$, is defined as:

$$V_{corr}=V_d·F_V$$

(17)

The determination of $F_V$ followed the same semi-empirical procedure used for $F_T$. The first step was to calculate the “target” value $F_{v,exp}$ for each experimental point by inverting Equation (17):

$$F_{v,exp}={\displaystyle \frac{V_{d,exp}}{V_d}}$$

(18)

After isolating the $F_{v,exp}$ values, the modeling challenge was addressed. A simple power law would not have been applicable due to the critical issue of the self-field regime, as the $\log \left(B_a\right)$ term is not defined for $B_a=0$.

To overcome this issue, a regression technique employing a unified linear model was adopted, based on the creation of two auxiliary variables:

• An indicator variable, ${\alpha }_{AF}$, defined as:

  $${\alpha }_{AF}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{B}_a>0(\mathrm{Applied-Field}\mathrm{regime})\hfill \\ 0\hfill & \mathrm{if}{B}_a=0(\mathrm{Self-Field}\mathrm{regime})\hfill \end{array}\right.$$
• A safe logarithmic variable, ${\left(\log \left(B_a\right)\right)}^{*}$, which contains $\log \left(B_a\right)$ only if $B_a>0$, and is 0 otherwise, ensuring numerical stability.

The complete linear model, provided to the fitlm regression function in MATLAB^® [19], was then structured in the logarithmic domain as:

$$\log \left(F_V\right)=\log \left(C_V\right)+{\alpha }_V\log \left(I_d\right)+{\beta }_V\log \left(\dot{m}\right)+{\delta }_V\log \left(r_a\right)+{ϵ}_V{\alpha }_{AF}+{\gamma }_V{\left(\log \left(B_a\right)\right)}^{*}$$

(19)

The effectiveness of this formulation lies in its ability to adapt to the operating context:

• In the Self-Field regime ($B_a=0$): ${\alpha }_{AF}=0$ and ${\left(\log \left(B_a\right)\right)}^{*}=0$, thus removing the corresponding terms and simplifying the equation to a reduced form.
• In the Applied-Field regime ($B_a>0$): ${\alpha }_{AF}=1$, activating the ${ϵ}_V$ term, which accounts for the transition effect, while ${\gamma }_V\log \left(B_a\right)$ models the dependence on the magnitude of the applied field.

Once the regression was performed and the coefficients determined, a piecewise function in the non-linear domain was obtained. After normalization with respect to the same reference values used in the thrust model ($I_{d,ref}=100$ A, ${\dot{m}}_{ref}=10$ mg/s, $B_{a,ref}=0.1$ T), plus $r_{a,ref}=15$ mm, the corrective factor $F_V$ takes the final form:

$$F_V=\left\{\begin{array}{cc}{C}_V·e^{{ϵ}_V}·{\left({\displaystyle \frac{I_d}{I_{d,ref}}}\right)}^{{\alpha }_V}·{\left({\displaystyle \frac{\dot{m}}{{\dot{m}}_{ref}}}\right)}^{{\beta }_V}·{\left({\displaystyle \frac{B_a}{B_{a,ref}}}\right)}^{{\gamma }_V}·{\left({\displaystyle \frac{r_a}{r_{a,ref}}}\right)}^{{\delta }_V},\hfill & \mathrm{if}{B}_a>0\hfill \\ C_V·{\left({\displaystyle \frac{I_d}{I_{d,ref}}}\right)}^{{\alpha }_V}·{\left({\displaystyle \frac{\dot{m}}{{\dot{m}}_{ref}}}\right)}^{{\beta }_V}·{\left({\displaystyle \frac{r_a}{r_{a,ref}}}\right)}^{{\delta }_V},\hfill & \mathrm{if}{B}_a=0\hfill \end{array}\right.$$

(20)

where the regression provided the dimensionless coefficients reported in

Table 3. Voltage model coefficients and validation.

	${\mathit{C}}_{\mathit{V}}$	${\mathit{ϵ}}_{\mathit{V}}$	${\mathit{\alpha }}_{\mathit{V}}$	${\mathit{\beta }}_{\mathit{V}}$	${\mathit{\gamma }}_{\mathit{V}}$	${\mathit{\delta }}_{\mathit{V}}$
Coefficient	0.77	0.75	0.13	0.02	0.07	0.50
Valid. Coeff.	0.71	0.80	0.12	0.06	0.10	0.53

as follows. As done before for the thrust model, also in the voltage model, the robustness and the sensitivity of the coefficients are evaluated by adding other two thruster configurations to the dataset, generating a new set of coefficients that in this case is slightly altered, but still demonstrates its reliability, as can be seen in the second row of Table 3.

This new formulation replaces Balkenhohl’s approach [6], offering a more robust and accurate voltage model [5,11,12], valid across both operating modes of the thruster.

6. Results

The main results obtained with their validation and comparison respect to the literature are reported here.

Table 4. Comparison between experimental thrust, the Balkenhohl model ($T_{Bal}$) [6] and the corrected model ($T_{corr}$). In each row, the lower error is highlighted in green and the higher one in red. Note that for those cases where $Ba=0$, the result is the same.

Thrust Configuration	${\mathit{B}}_{\mathit{a}}$ [T]	${\mathit{I}}_{\mathit{d}}$ [A]	${\mathit{T}}_{\mathit{exp}}$ [mN]	${\mathit{T}}_{\mathit{Bal}}$ [mN]	${\mathit{T}}_{\mathit{corr}}$ [mN]	Error % ${\mathit{T}}_{\mathit{Bal}}$	Error % ${\mathit{T}}_{\mathit{corr}}$
Beihang Univ. [17]	0	100	139	144.98	144.98	4.30	4.30
SF-Configuration (21 mg/s)	0	120	143	146.74	146.74	2.62	2.62
	0	150	149	149.98	149.98	0.66	0.66
	0	180	154	153.95	153.95	0.03	0.03
Waseda Univ. ($\varphi$4) [15] (3 mg/s)	0.15	15	6.5	23.91	16.48	267.81	153.58
	0.15	25	12.5	29.28	17.25	134.20	38.00
	0.15	35	14.0	34.71	18.32	147.90	30.84
DFVLR [14] (7 mg/s)	0.6	80	251	127.76	247.59	49.10	1.36
Beihang Univ. [17]	0.133	100	196	165.34	187.50	15.64	4.34
AF-Configuration (21 mg/s)	0.133	120	224	176.29	215.49	21.30	3.80
	0.133	150	259	193.20	264.76	25.41	2.23
	0.133	180	296	210.69	322.42	28.82	8.93
Beihang Univ. [16]	0.09	88	150	161.81	156.81	7.87	4.54
(21 mg/s)	0.09	120	195	177.81	182.72	8.81	6.30
	0.09	150	215	193.54	212.61	9.98	1.11
	0.09	180	250	209.96	247.67	16.02	0.93
Waseda Univ. ($\varphi$6) [15]	0.15	25	10.9	28.71	17.20	163.44	57.79
(3.0 mg/s) (Validation)	0.15	36.5	14.0	34.70	18.40	147.89	31.44

and

Table 5. Comparison between experimental discharge voltage, the Balkenhohl model ($V_{Bal}$) [6] and the corrected model ($V_{corr}$), with relative errors. In each row, the lower error is highlighted in green and the higher one in red.

Thrust Configuration	${\mathit{B}}_{\mathit{a}}$ [T]	${\mathit{I}}_{\mathit{d}}$ [A]	${\mathit{V}}_{\mathit{exp}}$ [V]	${\mathit{V}}_{\mathit{Bal}}$ [V]	${\mathit{V}}_{\mathit{corr}}$ [V]	Error % ${\mathit{V}}_{\mathit{Bal}}$	Error % ${\mathit{V}}_{\mathit{corr}}$
Beihang Univ. [17]	0	100	24.3	42.84	24.06	76.28	1.01
SF-Configuration (21 mg/s)	0	120	22.8	39.48	22.72	73.16	0.36
	0	150	21.3	36.25	21.50	70.20	0.92
	0	180	20.7	34.22	20.79	65.33	0.46
Waseda Univ. ($\varphi$4) [15] (3 mg/s)	0.15	15	31.5	45.09	30.86	43.16	2.03
	0.15	25	28.0	39.72	28.02	41.85	0.06
	0.15	35	27.0	38.36	27.54	42.08	2.00
DFVLR [14] (7 mg/s)	0.6	80	145.0	66.51	145.0	54.13	0.00
Beihang Univ. [17]	0.133	100	54.0	49.46	54.09	8.40	0.17
AF-Configuration (21 mg/s)	0.133	120	59.4	47.58	55.50	19.91	6.57
	0.133	150	59.2	46.35	59.70	21.70	0.84
	0.133	180	62.2	46.20	65.91	25.73	5.96
Beihang Univ. [16]	0.09	88	50.0	59.98	55.59	19.96	11.17
(21 mg/s)	0.09	120	57.0	54.67	54.27	4.08	4.79
	0.09	150	58.0	52.53	55.52	9.42	4.28
	0.09	180	59.0	51.74	58.24	12.31	1.30
Waseda Univ. ($\varphi$6) [15]	0.15	25	27.10	38.36	27.24	41.57	0.51
(3.0 mg/s) (Validation)	0.15	36.5	29.60	36.91	26.75	24.71	9.64

clearly demonstrate the significant improvement in predictive accuracy achieved by the new model.

Validation and Comparison with Experimental Data and Reference Model

Starting from the semi-empirical correction factor $F_T$ determined in the previous sections, the next crucial step was to validate the performance of the improved thrust model ($T_{corr}$). The primary objective of this validation is to quantify the model’s predictive accuracy and demonstrate its enhanced accuracy over the baseline Balkenhohl models.

The validation was performed by comparing the thrust predictions from both models against the experimental measurements ($T_{exp}$), which serve as the ground truth. For each operating point in the experimental dataset, the following values were calculated:

• Balkenhohl Thrust Model ($T_{Bal}$): The thrust predicted by the original reference model.
• Improved Thrust Model ($T_{corr}$): The thrust predicted by the improved model, incorporating the multi-variable correction factor $F_T$.

To provide a quantitative measure of accuracy, the percentage error for each prediction was calculated respect to the experimental value:

$$\mathrm{Error}[\%]={\displaystyle \frac{|T_{\mathrm{corr}}-T_{exp}|}{T_{exp}}}\times 100$$

(21)

The results of this comparative analysis are summarized in Table 4.

• In the self-field regime ($B_a=0$), where the $T_{af}$ component is null, the $F_T$ factor is irrelevant. As expected, the improved model’s predictions are identical to those of the Balkenhohl model. Both models show very good agreement with the experimental data for the SF-configuration of Beihang Univ. [17], with a maximum error of only 4.3%, confirming the validity of the underlying $T_{gd}$ and $T_{sf}$ formulations.
• In the applied-field regime ($B_a>0$), the advantages of the multi-variable correction become evident. The most critical case is that of the configuration of Waseda Univ. [15] (low mass flow rate, high magnetic field), where the Balkenhohl model [6] showed its greatest limitation, with errors exceeding 267%. The improved model, $T_{corr}$, drastically reduces this discrepancy, lowering the maximum error to about 154%.
• For more standard operating configurations, such as the AF configuration of Beihang Univ. (2019) [17] and the configuration of Beihang Univ. (2018) [16], a consistent improvement is observed across the entire current range. The Balkenhohl model exhibited errors between 15% and 29%, whereas the new model consistently keeps the error below 9%, often falling below 5%, with a maximum of 6.3% in the configuration of Beihang Univ. (2018) [16].

The last two rows report the results obtained by the validation of the thrust model on a couple of datasets not involved in the coefficient regression, demonstrating its strong reliability.

Following the development of the improved voltage model ($V_{corr}$), validation was conducted to quantify its predictive accuracy. As for the thrust, the performance of the new model was systematically compared against both the experimental measurements ($V_{d,exp}$) and the predictions from the baseline Balkenhohl model.

For each operating point in the experimental dataset, the percentage error was calculated for both models relative to the measured voltage:

$$\mathrm{Error}[\%]={\displaystyle \frac{|V_{\mathrm{corr}}-V_{exp}|}{V_{exp}}}\times 100$$

The comprehensive results of this comparative analysis are presented in Table 5. An examination of these results reveals a substantial and consistent improvement in accuracy provided by the new model.

• In the Self-Field Regime ($B_a=0$): This regime exposes the most significant weakness of the Balkenhohl model [6], which produces errors exceeding 70%. This is because its correction factor is not designed for zero-field operation. In stark contrast, the improved $V_{corr}$ model demonstrates exceptional accuracy, with prediction errors consistently around 1% or less. This confirms the success of the regime-dependent, piecewise structure of the new correction factor $F_V$.
• In the Applied-Field Regime ($B_a>0$): The improved model consistently outperforms the reference model. For the configuration of the Waseda Univ. [15], the error is reduced from a range of 42–43% down to approximately 2%. A similar trend is observed for the AF configuration of Beihang Univ. (2019) [17], where errors are reduced from an 8–26% range to below 6.6% in most cases.
• Robustness in Critical Operating Regimes: The model’s robustness is particularly evident in two specific cases. For the configuration of DFVLR (1975) [14], operating at a very high magnetic field, the new model achieves a near-perfect prediction with 0% error, a significant improvement over the 54% error of the baseline model. Furthermore, for the configuration of Beihang Univ. (2018) [16] at 180 A, where the Balkenhohl model [6] shows a 12.3% error, the new $V_{corr}$ model remains exceptionally accurate, with an error of just 1.3%. It is worth noting that only in one operating point (Beihang 2018, $I_d=120$ A), the corrected model performs slightly worse (4.8% vs. 4.1%), but the discrepancy is marginal and does not affect the overall improvement trend.

Also in this case, the last two rows report the results obtained by the validation of the voltage model on a couple of datasets not involved in the coefficient regression, demonstrating its strong reliability.

7. Discussion

The performance model for thrust and discharge voltage was developed and validated. The key innovation lies in the introduction of multi-factorial correction factors that simultaneously account for discharge current, mass flow rate, and the applied magnetic field.

Despite the relevance of the obtained results, it is worth noting that the model is currently limited to the power range below 12 kW and is associated only with argon-fed thrusters.

Nevertheless, a comparative analysis against experimental data and state-of-the-art reference models demonstrated a significant reduction in prediction error, typically from the 30–40% range to less than 6–9%. In particular, the physically consistent link between the corrected thrust $T_{corr}$ and the corrected discharge voltage $V_{corr}$ estimate enabled exceptionally accurate discharge voltage predictions, especially in the self-field and high applied-field regimes, where previous models showed the greatest limitations.

This capability paves the way for the development of a preliminary design tool capable of sizing and optimizing the thruster with specific parameters and performances.

Future research should be aimed at filling the gap related to the possibility of extending the model to different propellants and making it more generalized in terms of a wider power range.

8. Conclusions

Magnetoplasmadynamic thrusters seem to be one of the most promising electric propulsion systems for future space missions. Although this technology has always been considered for high-power applications, in the last few decades it has gained interest also in the low-power regime, yet its optimization is hampered by the lack of accurate performance models. Traditional formulations found in the literature prove inadequate to estimate thruster performances and fit the experimental data, especially in the presence of reduced propellant mass flow rate (3–21 mg/s) and strong magnetic fields (up to 0.6 T). The multifactorial correction approach, developed in this work and applied to both the thrust and the voltage estimation, has been proven to widely fill this gap with errors lower than 6–9%.

The validation confirms that the targeted correction of the $T_{af}$ component via the multi-variable factor $F_T$ effectively addresses the main weaknesses of the reference model. The resulting $T_{corr}$ model proves to be a more robust and accurate predictive tool, especially in the applied-field operational domain, which is of primary interest for this research.

For the new voltage model, the validation process confirms that it represents a major advancement. By establishing a more physically consistent base and developing a specific, multi-variable correction factor, the $V_{corr}$ model provides predictions that are not only significantly more accurate but also more robust across different operating ranges of the thruster.